LHPH101 (1) PDF Past Paper 2024-25
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2024
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This document appears to be a set of notes or a chapter on quantum physics, including information about quantum mechanics. It includes details about quantum concepts and calculations. It uses mathematical formulas and expressions.
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vè;k; 1 oS|qr vkos'k rFkk {ks=k 1.1 Hkwfedk ge lHkh dks] fo'ks"kdj 'kq"d ekSle esa] LosVj vFkok laf'y"V oL=kksa dks 'kjhj ls mrkjrs le; pV&pV dh èofu lquus vFkok fpuxkfj;k¡ ns[kus dk vuqHko gksxkA D;k vkius dHkh bl ifj?kVuk dk Li"Vhdj.k [kkstus dk iz;kl fd;k gS\ fo|qr foltZu dk ,d vU; lkekU; mnkgj...
vè;k; 1 oS|qr vkos'k rFkk {ks=k 1.1 Hkwfedk ge lHkh dks] fo'ks"kdj 'kq"d ekSle esa] LosVj vFkok laf'y"V oL=kksa dks 'kjhj ls mrkjrs le; pV&pV dh èofu lquus vFkok fpuxkfj;k¡ ns[kus dk vuqHko gksxkA D;k vkius dHkh bl ifj?kVuk dk Li"Vhdj.k [kkstus dk iz;kl fd;k gS\ fo|qr foltZu dk ,d vU; lkekU; mnkgj.k vkdk'k esa xtZu osQ le; rfM+r fn[kkbZ nsuk gSA fo|qr >VosQ osQ laosnu dk vuqHko gesa ml le; Hkh gksrk gS tc ge fdlh dkj dk njokTk+ k [kksyrs gSa vFkok tc ge viuh cl dh lhV ij f[kldus osQ i'pkr mlesa yxh yksgs dh NM+ dks idM+rs gSaA bu vuqHkoksa osQ gksus osQ dkj.k gekjs 'kjhj esa ls gksdj mu oS|qr vkos'kksa dk fol£tr gksuk gS tks fo|qrjks/h i`"Bksa ij jxM+ osQ dkj.k ,d=k gks tkrs gSaA vkius ;g Hkh lquk gksxk fd ;g oS|qr vkos'k (fLFkjoS|qr) osQ mRiUu gksus osQ dkj.k gSA bl vè;k; rFkk vxys vè;k; esa Hkh ge blh fo"k; ij ppkZ djsaxsA fLFkj ls rkRi;Z gS og lc oqQN tks le; osQ lkFk ifjo£rr vFkok xfre; ugha gksrkA fLFkjoS|fq rdh osQ varxZr ge fLFkj vkos'kksa }kjk mRiUu cyksa] {ks=kksa rFkk foHkoksa osQ fo"k; esa vè;;u djrs gSaA 1.2 oS|qr vkos'k bfrgkl osQ vuqlkj yxHkx 600 bZ- iwoZ gqbZ bl rF; dh [kkst dk Js;] fd Åu vFkok js'keh&oL=k ls jxM+k x;k ,sEcj gydh oLrqvksa dks vkd£"kr djrk gS] xzhl ns'k osQ feysVl osQ fuoklh FksYl dks tkrk gSA ^bysfDVªflVh* 'kCn Hkh xzhl dh Hkk"kk osQ 'kCn bysDVªkWu ls O;qRiUu gqvk gS ftldk vFkZ ,sEcj gSA ml le; inkFkks± osQ ,sls cgqr ls ;qxy Kkr Fks tks ijLij jxM+s 2024-25 HkkSfrdh tkus ij Hkwls osQ frudksa] ljoaQMs dh xksfy;ksa] dkxTk+ osQ NksVs VqdM+ksa vkfn gydh oLrqvksa dks vkd£"kr dj ysrs FksA ;g Hkh izsf{kr fd;k x;k fd ;fn Åu vFkok js'ke osQ diM+s ls jxM+h gqbZ nks dk¡p dh NM+ksa dks ,d&nwljs osQ fudV yk,¡ rks os ,d&nwljs dks izfrd£"kr djrh gSa ¹fp=k 1.1(a)ºA Åu dh os yfM+;k¡ vFkok js'ke osQ diM+s osQ os VqdM+s ftuls bu NM+ksa dks jxM+k fp=k 1.1 NM+sa% ltkrh; vkos'k ,d&nwljs dks izfrd£"kr rFkk x;k Fkk] os Hkh ijLij ,d&nwljs dks izfrd£"kr djrs fotkrh; vkos'k ,d&nwljs dks vkd£"kr djrs gSaA gSa ijarq dk¡p dh NM+ rFkk Åu ,d&nwljs dks vkd£"kr djrs gSaA blh izdkj] fcYyh dh lewj ls jxM+h gqbZ nks IykfLVd dh NM+sa ,d&nwljs dks izfrd£"kr djrh gSa ¹fp=k 1.1(b)º ijarq lewj dks vkd£"kr djrh gSaA blosQ foijhr] IykfLVd dh NM+sa dk¡p dh NM+ksa dks vkd£"kr djrh gSa ¹fp=k 1.1(c)º rFkk flYd vFkok Åu ftlls dk¡p dh NM+ksa dks jxM+k x;k Fkk] dks izfrd£"kr djrh gSaA dk¡p dh NM+ lewj dks izfrd£"kr djrh gSA o"kks± osQ iz;kl rFkk lko/kuhiwod Z fd, x, iz;ksxksa ,oa muosQ fo'ys"k.kksa }kjk ljy izrhr gksus okys ;s rF; LFkkfir gks ik, gSaA fofHkUu oSKkfudksa }kjk fd, x, cgqr ls lko/kuhiw.kZ vè;;uksa osQ i'pkr ;g fu"d"kZ fudkyk x;k gS fd ,d jkf'k gksrh gS] ftls oS|qr vkos'k dgrs gSa vkSj ;g osQoy nks izdkj osQ gh gks ldrs gSaA oS|qr vkos'k dgykus okyh jkf'k osQ osQoy nks izdkj gh gksrs gSaA ge dgrs gSa fd IykfLVd ,oa dk¡p dh NM+] js'ke] lewj] ljoaQMs dh xksfy;k¡ vkfn ¯iM fo|qUe; gks x, gSaA jxM+us ij ;s oS|qr vkos'k v£tr dj ysrs gSaA vkos'k nks izdkj osQ gksrs gSa rFkk ge ;g ikrs gSa fd (i) ltkrh; vkos'k ,d&nwljs dks izfrd£"kr rFkk (ii) fotkrh; vkos'k ,d&nwljs dks vkd£"kr djrs gSaA og xq.k tks nks izdkj osQ vkos'kksa esa Hksn djrk gS] vkos'k dh /qzork dgykrk gSA tc dk¡p dh NM+ dks js'ke ls jxM+rs gSa rks NM+ ,d izdkj dk vkos'k v£tr djrh gS rFkk js'ke nwljs izdkj dk vkos'k v£tr djrk gSA ;g mu lHkh oLrqvksa osQ ;qxy osQ fy, lR; gS tks fo|qUe; gksus osQ fy, ijLij jxM+s tkrs gSaA vc ;fn fo|qUe; dk¡p dh NM+ dks ml js'ke osQ laioZQ esa ykrs gSa ftlls mls jxM+k x;k Fkk] rks os vc ,d&nwljs dks vkd£"kr ugha djrsA ;s vc vU; gydh oLrqvksa dks Hkh vkd£"kr vFkok izfrd£"kr ugha djrs tSlk fd ;s fo|qUe; gksus ij dj jgs FksA bl izdkj] jxM+us osQ i'pkr oLrqvksa }kjk v£tr vkos'k vkosf'kr oLrqvksa dks ,d&nwljs osQ laioZQ esa ykus ij yqIr gks tkrk gSA bu iz{s k.kksa ls vki D;k fu"d"kZ fudky ldrs gS\a ;g rks osQoy bruk crkrk gS fd oLrqvksa }kjk v£tr fotkrh; vkos'k ,d&nwljs osQ izHkko dks fu"iQy dj nsrs gSaA blhfy, vesfjdh oSKkfud csatkfeu izsaQdfyu us vkos'kksa dks /ukRed rFkk ½.kkRed dgkA ifjikVh osQ vuqlkj dk¡p dh NM+ vFkok fcYyh osQ lewj ij vkos'k /ukRed dgykrk gS rFkk IykfLVd&NM+ vFkok js'ke ij vkos'k ½.kkRed dgykrk gSA tc fdlh oLrq ij dksbZ vkos'k gksrk gS rks og oLrq fo|qUe; vFkok vkosf'kr (vkfo"V) dgh tkrh gSA tc ml ij dksbZ vkos'k ugha gksrk rc mls vukosf'kr dgrs gSaA vkos'kksa dh mifLFkfr osQ lalwpu osQ fy, ,d ljy midj.k Lo.kZ i=k fo|qrn'khZ gS HkkSfrdh ¹fp=k 1.2 (a)ºA blesa ,d ckWDl esa /krq dh ,d NM+ ÅèokZèkjr% yxh gksrh gS ftlosQ fupys fljs ij lksus osQ ooZQ dh nks ifêð;k¡ c¡/h gksrh gSaA tc dksbZ vkosf'kr oLrq NM+ osQ Åijh fljs dks Nwrh gS rks NM+ esa gksrk gqvk vkos'k lksus osQ odks± ij vk tkrk gS vkSj os ,d&nwljs ls nwj gV tkrs gSaA vkos'k ftruk vf/d gksrk gS] odks± osQ fupys fljksa osQ chp mruh gh vf/d nwjh 2 gks tkrh gSA 2024-25 oS|qr vkos'k rFkk {ks=k vkb,] vc ;g le>sa fd nzO; ls cuh oLrq,¡ D;ksa vkos'k dks v£tr djrh gSaA lHkh inkFkZ ijek.kqvksa vkSj@vFkok v.kqvksa ls cus gSAa ;|fi oLrq,¡ lkekU;r% oS|qr mnklhu gksrh gSa] muesa vkos'k rks gksrs gSa ijarq muosQ ;s vkos'k Bhd&Bhd larqfyr gksrs gSaA v.kqvksa dks l¡Hkkyus okyk jklk;fud cy] Bkslksa esa ijek.kqvksa dks ,dlkFk Fkkes j[kus okys cy] xksan dk vklatd cy] i`"B ruko ls lac¼ cyµbu lHkh cyksa dh ewy izo`Qfr oS|qrh; gS] vkSj ;s vkosf'kr d.kksa osQ chp yxus okys fo|qr cyksa ls mRiUu gksrs gSaA bl izdkj] oS|qrpqacdh; cy loZO;kih gS vkSj ;g gekjs thou ls lac¼ izR;sd {ks=k esa lfEefyr gSA vr% ;g vko';d gS fd ge bl izdkj osQ cy osQ fo"k; esa vf/d tkudkjh izkIr djsaA fdlh mnklhu oLrq dks vkosf'kr djus osQ fy, gesa mlls ,d izdkj osQ vkos'k dks tksMu+ s vFkok gVkus dh vko';drk gksrh gSA tc ge ;g dgrs gSa fd dksbZ oLrq vkosf'kr gS rks ge lnSo gh bl vkos'k osQ vkf/D; vFkok vHkko dk mYys[k djrs gSaA Bkslksa esa oqQN bysDVªkuW ijek.kq esa de dldj vkc¼ gksus osQ dkj.k] os vkos'k gksrs gSa tks ,d oLrq ls nwljh oLrq esa LFkkukarfjr gks tkrs gSaA bl izdkj dksbZ oLrq vius oqQN bysDVªkWu [kksdj /ukosf'kr gks ldrh gSA blh izdkj fdlh oLrq dks bysDVªkWu nsdj ½.kkosf'kr Hkh cuk;k tk ldrk fp=k 1.2 fo|qrn'khZ (a) Lo.kZ i=k fo|qrn'khZ (b) ljy fo|qrn'khZ dh :ijs[kkA gSA tc ge dk¡p dh NM+ dks js'ke ls jxM+rs gSa rks NM+ osQ oqQN bysDVªkWu js'ke osQ diM+s esa LFkkukarfjr gks tkrs gSaA bl izdkj NM+ èkukosf'kr rFkk js'ke ½.kkosf'kr gks tkrk gSA jxM+us dh izfØ;k esa dksbZ u;k vkos'k mRiUu ugha gksrkA lkFk gh LFkkukarfjr gksus okys bysDVªkWuksa dh la[;k oLrq esa mifLFkr bysDVªkWuksa dh la[;k dh rqyuk esa ,d cgqr NksVk va'k gksrh gSA 1.3 pkyd rFkk fo|qrjks/h oqQN inkFkZ rqjar gh vius esa ls gksdj fo|qr dks izokfgr gksus nsrs gSa tcfd oqQN vU; ,slk ugha djrsA tks inkFkZ vklkuh ls vius esa ls gksdj fo|qr dks izokfgr gksus nsrs gSa mUgsa pkyd dgrs gSaA muesa ,sls oS|qr vkos'k (bysDVªkWu) gksrs gSa tks inkFkZ osQ Hkhrj xfr osQ fy, vis{kko`Qr Lora=k gksrs gSaA /krq,¡] ekuo rFkk tarq 'kjhj vkSj i`Foh pkyd gSaA dk¡p] ikWlZsysu] IykfLVd] ukW;yksu] ydM+h tSlh vf/dka'k v/krq,¡ vius ls gksdj izokfgr gksus okyh fo|qr ij mPp izfrjks/ yxkrh gSaA bUgsa fo|qrjks/h dgrs gSaA vf/dka'k inkFkZ Åij o£.kr bu nks oxks± esa ls fdlh ,d esa vkrs gSAa * tc oqQN vkos'k fdlh pkyd ij LFkkukarfjr gksrk gS rks og rqjar gh ml pkyd osQ leLr i`"B ij iSQy tkrk gSA blosQ foijhr ;fn oqQN vkos'k fdlh fo|qrjks/h dks nsa rks og ogha ij jgrk gSA ,slk D;ksa gksrk gS] ;g vki vxys vè;k; esa lh[ksaxsA * ,d rhljh Js.kh ftls v/Zpkyd dgrs gSa] vkos'kksa dh xfr esa vojks/ mRiUu djrh gSA bl vojks/ dk ifjekk.k pkydksa rFkk fo|qrjksf/;ksa osQ eè;orhZ gksrk gSA 3 2024-25 HkkSfrdh inkFkks± dk ;g xq.k geas crkrk gS fd lw[ks ckyksa esa oaQ?kh djus vFkok jxM+us ij ukW;yksu ;k IykfLVd dh oaQ?kh D;ksa vkosf'kr gks tkrh gS] ijarq /krq dh oLrq,¡ tSls pEep vkosf'kr D;ksa ugha gksrh\ /krqvksa ls vkos'k dk {kj.k gekjs 'kjhj ls gksdj /jrh esa gks tkrk gS] ,slk gksus dk dkj.k ;g gS fd /krq rFkk gekjk 'kjhj nksuksa gh fo|qr osQ vPNs pkyd gSaA ijarq ;fn /krq dh NM+ ij ydM+h vFkok IykfLVd dk gaSfMy yxk gS vkSj mlosQ èkkrq osQ Hkkx dks Li'kZ ugha fd;k x;k gS] rks og vkosf'kr gksus dk laosQr ns nsrh gSA 1.4 oS|qr vkos'k osQ ewy xq.k geus ;g ns[kk gS fd nks izdkj osQ vkos'k gksrs gSaµ /ukos'k rFkk ½.kkos'k rFkk buesa ,d&nwljs osQ izHkko dks fujLr djus dh izo`fÙk gksrh gSA vc] ge ;gk¡ oS|qr vkos'k osQ vU; xq.kksa dk o.kZu djsxa As ;fn vkosf'kr oLrqvksa dk lkb”k muosQ chp dh nwjh dh rqyuk esa cgqr de gksrk gS rks ge mUgsa ¯cnq vkos'k ekurs gSaA ;g eku fy;k tkrk gS fd oLrq dk laiw.kZ vkos'k vkdk'k esa ,d ¯cnq ij laosaQfnzr gSA 1.4.1 vkos'kksa dh ;ksT;rk vc rd geus vkos'k dh ifjek.kkRed ifjHkk"kk ugha nh gS_ bls ge vxys vuqHkkx esa le>saxsA varfje :i esa ge ;g ekusxa s fd ,slk fd;k tk ldrk gS vkSj fiQj vkxs cko loZizFke vaxzs”k iz;ksxdrkZ iSQjkMs }kjk [kksts x, fo|qr vi?kVu osQ izk;ksfxd fu;eksa ls izkIr gqvk FkkA lu~ 1912 esa fefydu us bls okLro esa izk;ksfxd :i ls fun£'kr fd;k FkkA ek=kdksa dh varjkZ"Vªh; iz.kkyh (SI) esa vkos'k dk ek=kd owQykWe gS] ftldk izrhd C gSA ,d owQykWe dks fo|qr /kjk osQ ek=kd osQ inksa esa ifjHkkf"kr fd;k tkrk gS ftlosQ fo"k; esa vki vxys vè;k; esa lh[ksaxsA bl ifjHkk"kk osQ vuqlkj] ,d owQykWe og vkos'k gS tks fdlh rkj esa 1 A (,ssfEi;j) /kjk 1 lsoaQM rd izokfgr djrk gS ¹HkkSfrdh dh ikB~;iqLrd d{kk 11] Hkkx 1 dk vè;k; 1 nsf[k,ºA bl iz.kkyh esa] vkos'k dh ewy bdkbZ e = 1.602192 × 10–19 C bl izdkj] –1C vkos'k esa yxHkx 6 × 1018 bysDVªkWu gksrs gSaA fLFkjoS|qfrdh esa brus fo'kky ifjek.k osQ vkos'kksa ls ;nk&dnk gh lkeuk gksrk gS vkSj blhfy, ge blosQ NksVs ek=kdksa 1 µC (ekbØksowQykWe) = 10–6 C vFkok 1 mC (feyhowQykWe) = 10–3 C dk mi;ksx djrs gSaA ;fn osQoy bysDVªkWu rFkk izksVkWu gh fo'o esa vkos'k osQ ewy ek=kd gSa rks lHkh izsf{kr vkos'kksa dks e dk iw.kk±d xq.kt gksuk pkfg,A bl izdkj ;fn fdlh oLrq esa n 1 bysDVªkWu rFkk n 2 izksVkWu gSa rks ml oLrq ij oqQy vkos'k n 2 × e + n1 × (–e) = (n2 – n1) e gSA pw¡fd n 1 rFkk n 2 iw.kk±d gSa] budk varj Hkh ,d iw.kk±d gSA vr% fdlh oLrq ij vkos'k lnSo e dk iw.kk±d xq.kt gksrk gS ftls e osQ pj.kksa esa gh ?kVk;k vFkok cqdk nsrs gSaA Li"V gS vc bl {ks=kiQy vo;o ls xqtjus okyh {ks=k js[kkvksa dh la[;k ?kV tk,xhA pw¡fd E osQ vfHkyacor {ks=kiQy vo;o ∆S dk iz{ksi ∆S cosθ gS] vr% ∆S ls xqtjus okyh {ks=k js[kkvksa dh la[;k E ∆S cosθ osQ vuqØekuqikrh gSA tc θ = 90° gksrk gS rks {ks=k js[kk,¡ ∆S osQ lekarj gks tkrh gSa vkSj blls dksbZ Hkh {ks=k js[kk ugha xqtjrh (fp=k 1.15 nsf[k,)A cgqr ls lanHkks± esa {ks=kiQy vo;o osQ ifjek.k osQ lkFk&lkFk mldk fno~QfoU;kl Hkh egRoiw.kZ gksrk gSA mnkgj.k osQ fy,] fdlh ty&izokg esa fdlh ¯jx ls xqtjus okys ty dk ifjek.k LokHkkfod :i ls bl ckr ij fuHkZj djrk gS fd vki ty /kjk esa bls fdl izdkj idM+s gq, gSaA ;fn vki bls ty&izokg osQ vfHkyacor j[krs gSa rks vU; lHkh fno~QfoU;klksa dh rqyuk esa bl foU;kl esa ¯jx ls vf/dre ty xqtjsxkA fp=k 1.15 E rFkk n̂ osQ chp >qdko θ ij blls ;g fu"d"kZ fudyrk gS fd {ks=kiQy&vo;o dks lfn'k ÝyDl dh fuHkZjrkA osQ leku ekuuk pkfg,A blesa ifjek.k osQ lkFk fn'kk Hkh gksrh gSA leryh; {ks=k dh fn'kk oSQls fu£n"V dh tk,\ Li"V :i ls ry ij vfHkyac ry dk fno~QfoU;kl fu£n"V djrk gSA bl izdkj leryh; {ks=k lfn'k dh fn'kk blosQ vfHkyac osQ vuqfn'k gksrh gSA fdlh ofØr i`"B osQ {ks=kiQy dks fdlh lfn'k ls oSQls lac¼ fd;k tkrk gS\ ge ;g dYiuk djrs gSa fd ofØr i`"B cgqr ls NksVs&NksVs {ks=kiQy vo;oksa esa foHkkftr gSA buesa ls izR;sd NksVk {ks=kiQy vo;o leryh; ekuk tk ldrk gS vkSj igys Li"Vhdj.k osQ vuqlkj blls lfn'k lac¼ fd;k tk ldrk gSA ;gk¡ ,d lafnX/rk ij è;ku nhft,A fdlh {ks=kiQy vo;o dh fn'kk mlosQ vfHkyac osQ vuqfn'k gksrh gSA ijarq vfHkyac nks fn'kk,¡ laosQr dj ldrk gSA fdlh {ks=kiQy vo;o ls lac¼ lfn'k dh fn'kk dk p;u fdl izdkj fd;k tkrk gS\ bl leL;k dk lek/ku bl lanHkZ esa mfpr oqQN ifjikfV;ksa osQ fu/kZj.k }kjk fd;k x;k gSA can i`"Bksa osQ izdj.kksa osQ fy, ;g ifjikVh vfr ljy gSA fdlh can i`"B osQ izR;sd {ks=kiQy vo;o ls lac¼ lfn'k dh fn'kk cfgeq[Z kh vfHkyac dh fn'kk ekuh tkrh gSA blh ifjikVh dk mi;ksx fp=k 1-16 esa fd;k x;k gSA bl izdkj] fdlh can i`"B osQ fdlh ¯cnq ij {ks=kiQy vo;o lfn'k ∆S dk eku ∆S n̂ gksrk gS] ;gk¡ ∆S {ks=kiQy lfn'k dk ifjek.k rFkk n̂ bl ¯cnq ij cfgeqZ[kh vfHkyac dh fn'kk esa ,dkad lfn'k gSA vc ge oS|qr ÝyDl dh ifjHkk"kk ij vkrs gSaA fdlh {ks=kiQy vo;o ∆S ls xqtjus okys oS|qr ÝyDl ∆φ dh ifjHkk"kk bl izdkj djrs gSa % ∆φ = E.∆S = E ∆S cosθ (1.11) tks igys dh Hkk¡fr bl {ks=kiQy vo;o dks dkVus okyh {ks=k js[kkvksa dh la[;k osQ vuqØekuqikrh fp=k 1.16 vfHkyac gSA ;gk¡ θ {ks=k vo;o ∆S rFkk E osQ chp dk dks.k gSA iwoksZDr ifjikVh osQ vuqlkj can i`"B osQ n̂ rFkk ∆S dks ifjHkkf"kr fy, θ {ks=k&vo;o ij cfgeqZ[kh vfHkyac rFkk E osQ chp dk dks.k gSA è;ku nhft,] ge O;atd djus dh ifjikVhA E ∆S cosθ ij nks > 2a) gS] rks q ,oa –q osQ dkj.k {ks=k yxHkx fujLr gks tkrs gSaA vr% vf/d nwfj;ksa ij fdlh oS|qr f}/qzo osQ dkj.k fo|qr {ks=k 1/r2 (,dy vkos'k q osQ dkj.k fo|qr {ks=k dh r ij fuHkZjrk) ls Hkh vf/d xfr ls ean gksrk tkrk gSA ;g xq.kkRed /kj.kk uhps fn, x, lqLi"V ifjdyu ls mRiUu gqbZ gS% 1.10.1 oS|qr f}/qzo osQ dkj.k {ks=k vkos'kksa osQ ;qxy (–q rFkk q) osQ dkj.k fno~QLFkku esa fdlh ¯cnq ij fo|qr {ks=k owQykWe fu;e rFkk vè;kjksi.k fl¼kar ls Kkr fd;k tk ldrk gSA fuEufyf[kr nks izdj.kksa osQ ifj.kke ljy gSa% (i) tc ¯cnq f}/qzo osQ v{k ij gS] (ii) tc ¯cnq f}/zqo osQ fo"kqorh; ry] vFkkZr f}/qzo v{k osQ osaQnz ls xqtjus okys f}/qzo v{k osQ yacor ry esa gSA fdlh O;kid ¯cnq P ij fo|qr {ks=k] vkos'k –q osQ dkj.k P ij fo|qr {ks=k E –q rFkk vkos'k +q osQ dkj.k P ij fo|qr {ks=k E+q dks lfn'kksa osQ lekarj prqHkqZt fu;e }kjk la;ksftr djosQ izkIr fd;k tkrk gSSA (i) v{k ij fLFkr ¯cnqvksa osQ fy, eku yhft, ¯cnq P f}/qzo osQ osaQnz ls q dh vksj fp=k (1.17a) esa n'kkZ, vuqlkj r nwjh ij gS] rc q E −q = − ˆ p [1.13(a)] 4 πε 0 (r + a )2 ;gk¡ p̂ f}/qzo v{k (–q ls q dh vksj) osQ vuqfn'k ,dkad lfn'k gSA lkFk gh q E +q = ˆ p [1.13(b)] 4 π ε 0 (r − a )2 P ij oqQy fo|qr {ks=k gS 23 2024-25 HkkSfrdh q 1 1 E = E+q + E−q = − p 4 π εo (r − a )2 (r + a )2 q 4a r = ˆ p (1.14) 4 π εo ( r 2 − a 2 )2 r >> a osQ fy, 4q a E= ˆ p (r >> a) (1.15) 4πεo r 3 (ii) fo"kqorh; ry ij fLFkr ¯cnqvksa osQ fy, nks vkos'kksa +q rFkk –q osQ dkj.k fo|qr {ks=kksa osQ ifjek.k q 1 E +q = [1.16(a)] 4 πε o r + a 2 2 q 1 E –q = [1.16(b)] 4 πε o r + a 2 2 leku gSaA fp=k 1.17 (a) v{k ij fLFkr fdlh ¯cnq] (b) f}/zqo E+q rFkk E–q dhfn'kk,¡ fp=k 1.17(b) esa n'kkZ;h xbZ gSAa Li"V gS fd osQ fo"kqorh; ry ij fLFkr fdlh ¯cnq ij f}/qzo dk f}/qoz v{k osQ vfHkyacor vo;o ,d&nwljs dks fujLr dj nsrs gSAa f}èkzoq fo|qr {ks=kA f}/qzo vk?kw.kZ p lfn'k gS ftldk ifjek.k v{k osQ vuqfn'k vo;o la;ksftr gks tkrs gSaA oqQy fo|qr {ks=k p̂ osQ p = q × 2a gS rFkk fn'kk foijhr gksrk gSA vr% –q ls q dh vksj gSA E = – (E +q + E –q ) cosθ p̂ 2qa =− ˆ p (1.17) 4 π ε o (r 2 + a 2 )3 / 2 vf/d nwfj;ksa (r >> a) ij 2qa E=− pˆ (r >> a ) (1.18) 4 π εo r 3 lehdj.kksa (1.15) rFkk (1.18) ls Li"V gS fd vf/d nwfj;ksa ij f}/qzo {ks=k esa q rFkk a i`Fkd :i ls lfEefyr ugha gksrs_ ;g buosQ la;qDr xq.kuiQy qa ij fuHkZj djrk gSA blls f}/qzo vk?kw.kZ dh ifjHkk"kk dk laosQr feyrk gSA fdlh oS|qr f}/qzo osQ f}/qzo vk?kw.kZ lfn'k p dh ifjHkk"kk bl izdkj dh tk ldrh gS% p = q × 2a p̂ (1.19) vFkkZr ;g ,d lfn'k gS ftldk ifjek.k vkos'k q rFkk i`Fkdu 2a (vkos'kksa q, –q osQ ;qxy osQ chp dh nwjh) rFkk fn'kk –q ls q dh vksj gksrh gSA p osQ inksa esa] fdlh f}/qzo dk fo|qr {ks=k vfèkd nwfj;ksa ij ,d ljy :i ys ysrk gSA f}/qzo v{k osQ fdlh ¯cnq ij 2p E= (r >> a) (1.20) 4πε or 3 fo"kqorh; ry osQ fdlh ¯cnq ij p E=− (r >> a) (1.21) 24 4πε or 3 2024-25 oS|qr vkos'k rFkk {ks=k bl egRoiw.kZ rF; ij è;ku nhft, fd f}/qzo {ks=k vf/d nwfj;ksa ij 1/r 2 osQ :i esa ugha oju~1/r 3 osQ :i esa ean gksrk gSA blosQ vfrfjDr f}/qzo {ks=k dk ifjek.k rFkk fn'kk osQoy nwjh r ij gh fuHkZj ugh g a S oju~ ;s lfn'k r rFkk f}/qoz vk?kw.kZ p osQ chp osQ dks.k ij Hkh fuHkZj djrs gSA ge mlosQ ckjs esa lksp ldrs gSa& tc f}/zqo vkeki 2a 'kwU; dh vksj vxzlj gksrk tkrk gS] rc vkos'k q vuar dh vksj vxzlj bl izdkj gksrk tkrk gS fd xq.kuiQy p = q × 2a ,d fu;r ifjfer la[;k gksrh gSA bl izdkj osQ f}/qzo dks ¯cnq f}/qzo dgrs gSaA fdlh ¯cnq f}/qzo osQ fy, lehdj.k (1.20) rFkk (1.21) r osQ lHkh ekuksa osQ fy, lR; rFkk ;FkkFkZ gSaA 1.10.2 f}/qzoksa dh HkkSfrd lkFkZdrk vf/dka'k v.kqvksa esa /ukos'kksa rFkk ½.kkos'kksa* osQ oasQnz ,d gh LFkku ij gksrs gSaA blhfy, buosQ f}/zqo vk?kw.kZ 'kwU; gksrs gSaA CO2 rFkk CH4 v.kq blh izdkj osQ gSaA fo|qr {ks=k vkjksfir fd, tkus ij ;s f}/qzo vk?kw.kZ fodflr dj ysrs gSa ijarq oqQN v.kqvksa eaas /ukos'kksa rFkk ½.kkos'kksa osQ osaQnz laikrh ugha gksrsA vr% fo|qr {ks=k dh vuqifLFkfr esa Hkh budk viuk LFkk;h f}/qzo vk?kw.kZ gksrk gSA bl izdkj osQ v.kqvksa dks /qzfor v.kq dgrs gSaA ty dk v.kq] H2O, bl izdkj osQ v.kqvksa dk ,d mnkgj.k gSA fofo/ inkFkZ fo|qr {ks=k dh mifLFkfr vFkok vuqifLFkfr esa jkspd xq.k rFkk egRoiw.kZ vuqiz;ksx izLrqr djrs gSaA mnkgj.k 1.9 ±10 µC osQ nks vkos'k ,d&nwljs ls 5.0 mm nwjh ij fLFkr gSaA (a) bl f}èkqzo osQ v{k ij f}/qzo osQ osaQnz O ls fp=k 1.18(a) esa n'kkZ, vuqlkj] /ukos'k dh vksj 15 cm nwjh ij fLFkr fdlh ¯cnq P ij rFkk (b) f}/qoz osQ v{k osQ vfHkyacor O ls] fp=k 1.18(b) esa n'kkZ, vuqlkj xqtjus okyh js[kk ls 15 cm nwjh ij fLFkr fdlh ¯cnq Q ij fo|qr {ks=k Kkr dhft,A mnkgj.k 1.9 fp=k 1.18 gy % (a) ¯cnq P ij vkos'k +10 µ C osQ dkj.k {ks=k * /ukRed ¯cnq vkos'kksa osQ laxzg dks osaQnz dh ifjHkk"kk lagfr osaQnz dh gh Hkk¡fr dh tkrh gS ftlosQ vuqlkj ∑ qi ri i rcm =. ∑ qi 25 i 2024-25 HkkSfrdh 10 −5 C 1 = × 4 π (8.854 × 10 −12 C 2 N −1 m −2 ) (15 − 0.25)2 × 10 −4 m 2 = 4.13 × 106 N C–1 BP osQ vuqfn'k ¯cnq P ij vkos'k –10 µC osQ dkj.k {ks=k 10 –5 C 1 = × 4 π (8.854 × 10 −12 C2 N −1 m −2 ) (15 + 0.25)2 × 10 −4 m 2 = 3.86 × 106 N C–1 PA osQ vuqfn'k A rFkk B ij fLFkr nks vkos'kksa osQ dkj.k P ij ifj.kkeh fo|qr {ks=k = 2.7 × 105 N C–1 BP osQ vuqfn'k gSA bl mnkgj.k esa vuqikr OP/OB dkiQh vf/d (=60) gSA vr%] fdlh f}/qzo osQ v{k ij vR;fèkd nwjh ij fLFkr fdlh ¯cnq ij fo|qr {ks=k Kkr djus osQ fy, lh/s gh lw=k osQ mi;ksx }kjk ge blh osQ lfUudV ifj.kke dh vk'kk dj ldrs gSaA 2a i`Fkdu osQ ± q vkos'kksa ls cus f}èkqzo osQ fy, f}/qzo osQ v{k osQ oasQnz ls r nwjh ij fo|qr {ks=k dk ifjek.k 2p E = (r/a >> 1) 4 π ε 0r 3 ;gk¡ p = 2a q f}/zqo vk?kw.kZ dk ifjek.k gSA f}/zqzo v{k ij fo|qr {ks=k dh fn'kk lnSo f}/qzo vk?kw.kZ lfn'k osQ vuqfn'k (vFkkZr –q ls q dh vksj) gksrh gSA ;gk¡] p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m vr% 2 × 5 × 10 −8 C m 1 E = × = 2.6 × 105 N C–1 4 π (8.854 × 10 −12 C2 N −1 m −2 ) (15)3 × 10 −6 m 3 f}/qzo vk?kw.kZ dh fn'kk AB osQ vuqfn'k gS] rFkk ;g ifj.kke iwoZ ifj.kke osQ dkiQh fudV gSA (b) ¯cnq B ij fLFkr + 10 µC vkos'k osQ dkj.k ¯cnq Q ij fo|qr {ks=k 10 −5 C 1 = 4 π (8.854 × 10 −12 C2 N −1 m −2 ) × [152 + (0.25)2 ] × 10 −4 m 2 = 3.99 × 106 N C–1 BQ osQ vuqfn'k ¯cnq A ij fLFkr –10 µC vkos'k osQ dkj.k Q ij fo|qr {ks=k 10 −5 C 1 = × 4 π (8.854 × 10 −12 C2 N −1 m −2 ) [152 + (0.25)2 ] × 10 −4 m 2 = 3.99 × 106 N C–1 QA osQ vuqfn'k Li"V gS fd bu nks leku ifjek.k osQ cyksa osQ OQ fn'kk osQ vuqfn'k ?kVd ,d&nwljs dks fujLr djrs gSa ijarq BA osQ lekarj fn'kk osQ vuqfn'k ?kVd la;ksftr gks tkrs gSaA vr% A rFkk B ij fLFkr nks vkos'kksa osQ dkj.k Q ij ifj.kkeh fo|qr {ks=k 0.25 =2× × 3.99 × 106 N C –1 BA osQ vuqfn'k 2 15 + (0.25)2 mnkgj.k 1.9 = 1.33 × 105 N C–1 BA osQ vuqfn'k (a) dh gh Hkk¡fr f}/qoz osQ v{k osQ vfHkyacor fdlh ¯cnq ij f}/qoz fo|qr {ks=k osQ fy, lh/s gh lw=k osQ mi;ksx }kjk ge blh ifj.kke dh vis{kk dj ldrs gSaµ 26 2024-25 oS|qr vkos'k rFkk {ks=k p E = (r/a >> 1) 4 π ε0 r 3 5 × 10−8 C m 1 = × −12 2 4 π (8.854 ×10 C N m )–1 –2 (15) × 10 −6 m 3 3 mnkgj.k 1.9 = 1.33 × 105 N C–1. bl izdj.k esa fo|qr {ks=k dh fn'kk vk?kw.kZ lfn'k dh fn'kk osQ foijhr gSA rFkkfi izkIr ifj.kke igys izkIr gq, ifj.kke osQ vuq:i gSaA 1.11 ,dleku cká {ks=k esa f}/zqo fp=k 1.19 esa n'kkZ, vuqlkj ,dleku fo|qr {ks=k E esa j[ks f}/qzo vk?kw.kZ p osQ LFkk;h f}èkqzo (LFkk;h f}èkqzo ls gekjk rkRi;Z ;g gS fd p dk E ls Lora=k vfLrRo gS_ ;g E }kjk izsfjr ugha gqvk gSA) ij fopkj dhft,A ;gk¡ vkos'k q ij qE rFkk –q ij –qE cy yx jgs gSaA pw¡fd E ,dleku gS vr% f}/zqo ij usV cy 'kwU; gSA ijarq vkos'kksa esa i`Fkdu gS] vr% cy fHkUu ¯cnq ij yxs gSa] ftlosQ ifj.kkeLo:i f}/qzo ij cy vk?kw.kZ dk;Z djrk gSA tc usV cy 'kwU; gS rks cy vk?kw.kZ (cy ;qXe) ewy ¯cnq ij fuHkZj ugha gksrkA bldk ifjek.k izR;sd cy osQ ifjek.k fp=k 1.19 ,dleku fo|qr {ks=k rFkk cy;qXe dh Hkqtk (nks izfrlekarj cyksa osQ chp yacor nwjh) osQ xq.kuiQy osQ cjkcj esa f}/qzoA gksrk gSA cy vk?kw.kZ dk ifjek.k = q E × 2 a sinθ = 2 q a E sinθ bldh fn'kk dkxT+k osQ ry osQ vfHkyacor blls ckgj dh vksj gSA p × E dk ifjek.k Hkh p E sinθ gS rFkk bldh fn'kk dkxT+k osQ i`"B osQ vfHkyacor ckgj dh vksj gSA vr% τ = p×E (1.22) ;g cy vk?kw.kZ f}/qzo dks {ks=k E osQ lkFk lajsf[kr djus dh izo`fÙk j[ksxkA tc p {ks=k E osQ lkFk lajsf[kr gks tkrk gS rks cy vk?kw.kZ 'kwU; gksrk gSA tc {ks=k ,dleku ugha gksrk rc D;k gksrk gS\ Li"V gS] ml izdj.k esa usV cy 'kwU;srj gks ldrk gSA blosQ vfrfjDr] O;kid :i ls fudk; ij igys dh gh Hkk¡fr ,d cy vk?kw.kZ dk;Z djsxkA ;gk¡ O;kid izdj.k varxzLZ r gS] vr% vkb, ,slh ljy fLFkfr ij fopkj djrs gSa ftlesa p {ks=k E osQ lekarj vFkok izfrlekarj gSA nksuksa gh izdj.kksa esa usV cy vk?kw.kZ rks 'kwU; gksrk gS ijarq ;fn E ,dleku ugha gS rks f}/qzo ij ,d usV cy yxrk gSA fp=k 1.20 Lor% Li"Vhdj.k djrk gSA bls vklkuh ls ns[kk tk ldrk gS fd tc p {ks=k E osQ lekarj gS rks f}/zqo ij c 0 gS rks A dh fn'kk ,dkad lfn'k â osQ leku gksxh rFkk ;fn A < 0 gS rks A dh fn'kk â dh fn'kk osQ foijhr gksxhA tc ge ½.ksrj ekuksa rd lhfer j[kuk pkgrs gSa rks ge izrhd A dk mi;ksx djrs gSa rFkk bls A dk ekM~;wYl (ekikad) dgrs gSaA bl izdkj A ≥ 0 gksrk gSA ;g Hkh è;ku nhft, fd mijksDr ppkZ esa ;|fi i`"B (λl ) }kjk ifjc¼ vkos'k dks gh osQoy lfEefyr fd;k x;k Fkk] ijarq fo|qr {ks=k E leLr rkj ij vkos'k osQ dkj.k gSA lkFk gh ;g dYiuk dj ysuk fd rkj dh yackbZ vuar gS bl dYiuk osQ fcuk ge fo|qr {ks=k E dks csyukdkj xkmlh; i`"B osQ vfHkyacor ugha ys ldrsA rFkkfi] yacs rkj osQ csyukdkj Hkkxksa osQ pkjksa vksj tgk¡ vaR; izHkko (end effects) dh mis{kk dh tk ldrh gS] fo|qr {ks=k osQ fy, lehdj.k (1.32) lfUudVr% lgh gSA 1.14.2 ,dleku vkosf'kr vuar lery pknj osQ dkj.k fo|qr {ks=k eku yhft, fdlh vuar lery pknj (fp=k 1.27) dk ,dleku i`"Bh; vkos'k ?kuRo σ gSA ge x-v{k dks fn, x, ry osQ vfHkyacor ekurs gSaA lefefr osQ vuqlkj fo|qr {ks=k y rFkk z funsZ'kkadksa ij fuHkZj ugha djsxk rFkk bldh izR;sd ¯cnq ij fn'kk x-fn'kk osQ lekarj gksuh pkfg,A ge xkmlh; i`"B dks fp=k esa n'kkZ, vuqlkj A vuqizLFkdkV {ks=kiQy osQ vk;rkdkj lekarj "kV~iQyd tSlk ys ldrs gSa (oSls rks csyukdkj i`"B ls Hkh ;g dk;Z gks ldrk gS)A tSlk fp=k ls n`f"Vxkspj gksrk gS] osQoy nks iQyd 1 rFkk 2 gh ÝyDl esa ;ksxnku nsaxs_ fo|qr {ks=k js[kk,¡ vU; iQydksa osQ lekarj gSa vkSj os blhfy, oqQy ÝyDl esa ;ksxnku ugha nsrhaA i`"B 1 osQ vfHkyacor ,dkad lfn'k –x fn'kk esa gS tcfd i`"B 2 osQ vfHkyacor ,dkad lfn'k +x fn'kk esa gSA vr%] nksuksa i`"Bksa ls xqtjus okys ÝyDl E.∆S cjkcj gSa vkSj la;ksftr gks tkrs gSaA fp=k 1.27 ,dleku vkosf'kr vuar vkos'k pknj osQ blhfy, xkmlh; i`"B ls xqtjus okyk usV ÝyDl 2 EA gSA i`"B }kjk fy, xkmlh; i`"BA ifjc¼ vkos'k σA gSA blhfy, xkml fu;e }kjk gesa ;g laca/ izkIr 34 gksrk gS 2024-25 oS|qr vkos'k rFkk {ks=k 2 EA = σA/ε 0 vFkok E = σ/2ε 0 lfn'k :i esa σ E= ˆ n (1.33) 2ε0 ;gk¡ n̂ ry osQ vfHkyacor blls nwj tkrk gqvk ,dkad lfn'k gSA ;fn σ /ukRed gS rks E ry ls cfgeZq[kh rFkk ;fn σ ½.kkRed gS rks E ry ls vareqZ[kh gksrk gSA è;ku nhft, xkml fu;e osQ vuqiz;ksx ls gesa ,d vfrfjDr rF; ;g izkIr gksrk gS fd E, x ij Hkh fuHkZj ugha gSA fdlh ifjfer cM+h leryh; pknj osQ fy, lehdj.k (1.33), fljksa ls nwj leryh; pknj osQ eè;orhZ {ks=kksa esa lfUudVr% lR; gSA 1.14.3 ,dleku vkosf'kr irys xksyh; [kksy ds dkj.k fo|qr {ks=k eku yhft, R f=kT;k ds irys xksyh; [kksy dk ,dleku i`"Bh; vkos'k ?kuRo σ gS (fp=k 1.28)A Li"V :i ls bl fLFkfr esa xksyh; lefefr gSA fdlh fcanq P ij pkgs og Hkhrj gS vFkok ckgj] fo|qr {ks=k osQoy r ij fuHkZj dj ldrk gS (;gk¡ r [kksy osQ osaQnz ls ml fcanq rd dh f=kT; nwjh gSA)rFkk bl {ks=k dks vjh; (vFkkZr èkzqokarj osQ vuqfn'k) gksuk pkfg,A (i) [kksy osQ ckgj fo|qr {ks=kµ[kksy osQ ckgj /zqokarj r osQ fdlh fcanq P ij fopkj dhft,A fcanq P ij E dk ifjdyu djus osQ fy, ge osaQnz O rFkk f=kT;k r osQ fcanq P ls xqtjus okys xksys dks xkmlh; i`"B ekurs gSaA fn, x, vkos'k foU;kl osQ lkis{k bl xksys ij fLFkr izR;sd fcanq lerqY; gSA (xksyh; lefefr ls gekjk ;gh vfHkizk; gSA) blhfy, xkmlh; i`"B osQ izR;sd fcanq ij fo|qr {ks=k dk leku ifjek.k E gS rFkk izR;sd fcanq ij /zqokarj osQ vuqfn'k gSA bl izdkj izR;sd fcanq ij E rFkk ∆S lekarj gSa rFkk izR;sd vo;o ls xq”kjus okyk ÝyDl E ∆S gSA lHkh ∆S dk ladyu djus ij xkmlh; i`"B ls xq”kjus okyk ÝyDl E × 4 π r 2 gSA ifjc¼ vkos'k σ × 4 π R 2 gSA xkml fu;e ls σ2 E × 4 π r2 = ε 4 π R 0 σ R2 q vFkok E= = ε0 r 2 4 π ε0 r 2 ;gk¡ q = 4 π R2 σ xksyh; [kksy ij oqQy vkos'k gSA q lfn'k :i esa] E = 4 πε r 2 rˆ (1.34) fp=k 1.28 fdlh ¯cnq osQ fy, tks 0 (a) r > R, (b) r < R ij gS] xkmlh; ;fn q > 0 gS rks fo|qr {ks=k cfgeqZ[kh gksrk gS rFkk ;fn q < 0 gS rks fo|qr {ks=k i`"BA vareqZ[kh gksrk gSA rFkkfi] ;g [kksy osQ osaQnz O ij fLFkr vkos'k q }kjk mRiUu fo|qr {ks=k gSA vr% [kksy osQ ckgj fLFkr fcanqvksa ij ,dleku vkosf'kr xksyh; [kksy osQ dkj.k fo|qr {ks=k bl izdkj dk gksrk gS] tSls fd [kksy dk leLr vkos'k mlosQ osQa nz ij fLFkr gSA (ii) [kksy osQ Hkhrj fo|qr {ks=kµfp=k 1.28(b) esa fcanq P [kksy osQ Hkhrj gSA bl izdj.k esa Hkh xkmlh; i`"B P ls xq”kjus okyk og xksyk gS ftldk osaQnz O gSA igys fd, x, 35 2024-25 HkkSfrdh ifjdyuksa dh gh Hkk¡fr xkmlh; i`"B ls xqtjus okyk ÝyDl E × 4 π r 2 gSA rFkkfi bl izdj.k esa] xkmlh; i`"B esa dksbZ vkos'k ifjc¼ ugha gSA rc xkml fu;e osQ vuqlkj E × 4 π r2 = 0 vFkkZr E = 0 (r < R ) (1.35) vFkkZr ,dleku vkosf'kr irys xksyh; [kksy* osQ dkj.k mlosQ Hkhrj fLFkr lHkh fcanqvksa ij fo|qr {ks=k 'kwU; gSA ;g egRoiw.kZ ifj.kke xkml fu;e dk izR;{k fu"d"kZ gS tks dwykWe fu;e ls izkIr gqvk gSA bl fu;e dk izk;ksfxd lR;kiu dwykWe fu;e esa 1/r2 dh fuHkZjrk dh iqf"V djrk gSA mnkgj.k 1.12 ijek.kq osQ izkjafHkd izfr:i esa ;g ekuk x;k Fkk fd vkos'k Ze dk fcanq vkeki dk /ukRed ukfHkd gksrk gS tks f=kT;k R rd ,dleku ?kuRo osQ ½.kkos'k ls f?kjk gqvk gSA ijek.kq iw.kZ :i esa fo|qr mnklhu gSA bl izfr:i ds fy, ukfHkd ls r nwjh ij fo|qr {ks=k fdruk gS\ fp=k 1-29 ijek.kq dk izkjafHkd izfr:iA gy% fp=k 1-29 esa bl izfr:i dk vkos'k forj.k n'kkZ;k x;k gSA pwf¡ d ijek.kq mnklhu (ukfHkd esa vkos'k Z e + ½.kkos'k) gS] vr% R f=kT;k osQ ,dleku xksyh; vkos'k forj.k esa oqQy ½.kkRed vkos'k –Ze gksuk pkfg,A blls gesa rqjar gh ½.kkRed vkos'k ?kuRo ρ izkIr gks tkrk gSA pw¡fd gekjs ikl dqy 'kwU; vkos'k gksuk pkfg,] vr% 4 πR3 ρ = 0 – Ze 3 3 Ze vFkok ρ = − 4 π R 3 ukfHkd ls r nwjh ij fLFkr fcanq P ij fo|qr {ks=k E(r) Kkr djus osQ fy, ge xkml fu;e dk mi;ksx djrs gSAa vkos'k forj.k dh xksyh; lefefr osQ dkj.k fo|qr {ks=k E(r) dk ifj.kke osQoy f=kT; nwjh ij fuHkZj djrk gS] ;gk¡ r dh fn'kk dk dksbZ vFkZ ugha gskrkA bldh fn'kk ewy fcanq ls P dh vksj /zqokarj r ds vuqfn'k (vFkok foijhr) gSA Li"V :i ls bl izdj.k esa xkmlh; i`"B ,d xksyh; i`"B gS ftldk osQa nz ukfHkd gSA ;gk¡ ge nks fLFkfr;ksa r < R rFkk r > R ij fopkj djrs gSaA (i) r < R : xksyh; i`"B }kjk ifjc¼ oS|qr ÝyDl mnkgj.k 1.12 φ = E (r ) × 4 π r 2 ;gk¡ E (r ), r ij fo|qr {ks=k dk ifjek.k gSA bldk dkj.k ;g gS xksyh; xkmlh; i`"B ds fdlh Hkh fcanq ij fo|qr {ks=k dh fn'kk i`"B osQ ml fcanq ij vfHkyacor gksrh gS rFkk bldk ifjek.k i`"B osQ lHkh fcanqvksa ij leku gksrk gSA * bldh rqyuk HkkSfrdh ikB~;iqLrd d{kk 11 osQ vuqHkkx 7-5 esa of.kZr ,dleku nzO;eku [kksy ls 36 dhft,A 2024-25 oS|qr vkos'k rFkk {ks=k ;gk¡ xkmlh; i`"B }kjk ifjc¼ vkos'k q /ukRed ukfHkdh; vkos'k rFkk r f=kT;k osQ xksys esa fo|eku ½.kkRed vkos'k gS 4 πr3 vFkkZr q = Ze + ρ 3 igys izkIr vkos'k ?kuRo ρ dk eku izfrLFkkfir djus ij gesa izkIr gksrk gSA r3 q = Ze−Ze R3 rc xkml fu;e ls gesa izkIr gksrk gSA Ze 1 r E (r ) = − ; r < R 4 π ε 0 r 2 R 3 ;gk¡ fo|qr {ks=k f=kT;r% cfgeqZ[kh funf'kZr gSA mnkgj.k 1.12 (ii) r > R : bl izdj.k esa] pw¡fd ijek.kq mnklhu gS] xksyh; xkmlh; i`"B }kjk ifjc¼ vkos'k 'kwU; gSA bl izdkj] xkml fu;e ls E (r) × 4 π r2 = 0 vFkok (r) = 0; r > R r = R, ij nksuksa izdj.kksa ls leku ifj.kke] E = 0 izkIr gksrk gSA lkjka'k 1. fo|qr rFkk pqacdh; cy ijek.kqvksa] v.kqvksa rFkk LFkwy nzO; osQ xq.kèkeks± dk fuèkkZj.k djrs gSaA 2. ?k"kZ.k fo|qr osQ ljy iz;ksxksa osQ vkèkkj ij ;g fu"d"kZ fudkyk tk ldrk gS fd izÑfr esa nks izdkj osQ vkos'k gksrs gSaA ltkrh; vkos'kksa esa izfrd"kZ.k rFkk fotkrh; vkos'kksa esa vkd"kZ.k gksrk gSA ifjikVh osQ vuqlkj] js'ke ls jxM+us ij dk¡p dh NM+ ij èkukos'k gksrk gS_ rFkk iQj ls jxM+us ij IykfLVd dh NM+ ij ½.kkos'k gksrk gSA 3. pkyd vius esa ls gksdj ljyrk ls oS|qr vkos'k dh xfr gksus nsrs gSa tcfd fo|qrjksèkh ,slk ugha djrsA èkkrqvksa esa xfr'khy vkos'k bysDVªkWu gksrs gSa_ oS|qr vi?kV~;ksa esa èkuk;u rFkk ½.kk;u nksuksa gh xfr djrs gSaA 4. oS|qr vkos'kksa osQ rhu xq.kèkeZ gksrs gSa % DokaVehdj.k] ;ksT;rk rFkk laj{k.kA oS|qr vkos'k osQ DokaVehdj.k ls rkRi;Z gS fd fdlh oLrq dk oqQy vkos'k (q) lnSo gh vkos'k osQ ,d ewy DokaVe (e) osQ iw.kk±dh xq.kt vFkkZr q = n e gksrk gS] tgk¡ n = 0, ±1, ±2, ±3,.... gSA izksVkWu rFkk bysDVªkWu ij Øe'k% +e rFkk –e vkos'k gksrs gSaA LFkwy vkos'kksa ftuosQ fy, n ,d vR;fèkd cM+h la[;k gksrh gS] esa vkos'k osQ DokaVehdj.k dh mis{kk dh tk ldrh gSA oS|qr vkos'kksa dh ;ksT;rk ls gekjk rkRi;Z ;g gS fd fdlh fudk; dk oqQy vkos'k ml fudk; osQ lHkh ,dkdh vkos'kksa dk chtxf.krh; ;ksx (vFkkZr ;ksx djrs le; muosQ fpÉksa dks è;ku esa j[kdj) gksrk gSA oS|qr vkos'kksa osQ laj{k.k ls gekjk rkRi;Z ;g gS fd fdlh fo;qDr fudk; (isolated system) dk oqQy vkos'k le; osQ lkFk vifjofrZr jgrk gSA bldk vFkZ ;g gS fd tc ?k"kZ.k }kjk oLrq,¡ vkosf'kr dh tkrh gSa rks vkos'kksa dk ,d oLrq ls nwljh oLrq esa LFkkukarj.k gksrk gS] ijarq bl izfØ;k esa u rks dksbZ vkos'k mRiUu gksrk gS vkSj u gh u"V gksrk gSA 37 2024-25 HkkSfrdh 5. owQykWe fu;e % nks ¯cnq vkos'kksa q 1 rFkk q2 osQ chp ikjLifjd fLFkj oS|qr cy] vkos'kksa osQ xq.kuiQy q1q2 osQ vuqØekuqikrh rFkk muosQ chp dh nwjh r21 osQ oxZ osQ O;qRØekuqikrh gksrk gSA xf.krh; :i esa k (q1q2 ) F21 = q2 ij q1 osQ dkj.k yxus okyk cy = 2 r̂21 r21 1 ;gk¡ r̂21 vkos'k q1 ls q2 dh fn'kk esa ,dakd lfn'k gS rFkk k = vkuqikfrdrk 4 πε 0 fLFkjkad gSA SI ek=kdksa es]a vkos'k dk ek=kd owQykWe gSA fu;rkad ε0 dk izk;ksfxd eku gS ε0 = 8.854 × 10–12 C2 N–1 m–2 k dk lfUudV eku k = 9 × 109 N m2 C–2 gksrk gSA 6. fdlh izksVkWu rFkk fdlh bysDVªkWu osQ chp oS|qr cy rFkk xq#Rokd"kZ.k cy dk vuqikr gS] k e2 ≅ 2.4 × 1039 G m em p 7. vè;kjksi.k fl¼kar % ;g fl¼kar bl xq.kèkeZ ij vkèkkfjr gS fd nks vkos'kksa osQ chp yxus okys vkd"khZ vFkok izfrd"khZ cy fdlh rhljs (vFkok vfèkd) vfrfjDr vkos'k dh mifLFkfr ls izHkkfor ugha gksrAs vkos'kksa q1, q2, q3... osQ fdlh lewg osQ fy, fdlh vkos'k (tSls q1) ij cy] q1 ij q2 osQ dkj.k cy] q 1 ij q3 osQ dkj.k cy vkfn&vkfn osQ lfn'k ;ksx osQ cjkcj gksrk gSA izR;sd ;qxy osQ fy, ;g cy igys of.kZr nks vkos'kksa osQ fy, owQykWe osQ fu;e }kjk gh O;Dr fd;k tkrk gSA 8. fdlh vkos'k foU;kl osQ dkj.k fdlh ¯cnq ij fo|qr {ks=k E fdlh NksVs èkukRed ijh{k.k vkos'k (test charge) q dks ml ¯cnq ij j[kus ij mlosQ }kjk vuqHko fd, tkus okys cy dks ml vkos'k osQ ifjek.k }kjk foHkkftr djus ij izkIr gksrk gSA fdlh ¯cnq vkos'k q osQ dkj.k fo|qr {ks=k dk dksbZ ifjek.k q / 4 π ε 0 r 2 gksrk gS_ ;fn q èkukRed gS rks ;g {ks=k vjh; (f=kT;h;) cfgeqZ[kh gksrk gS rFkk ;fn q ½.kkRed gS rks vjh; (f=kT;h;) vareqZ[kh gksrk gSA owQykWe cy dh Hkk¡fr fo|qr {ks=k Hkh vè;kjksi.k fl¼kar dks larq"V djrk gSA 9. fo|qr {ks=k js[kk ,slk oØ gS ftlosQ fdlh Hkh ¯cnq ij [khapk x;k Li'khZ] oØ osQ ml ¯cnq ij fo|qr {ks=k dh fn'kk crkrk gSA {ks=k js[kkvksa dh lkisf{kd ladqyrk fofHkUu fcanqvksa ij fo|qr {ks=k dh lkisf{kd rhozrk dks bafxr djrh gSA izcy fo|qr {ks=k esa js[kk,¡ ikl&ikl rFkk nqcZy {ks=k esa ;s ,d-nwljs ls dkiQh nwj gksrh gSaA ,dleku (vFkok fu;r) fo|qr {ks=k esa {ks=k js[kk,¡ ,d-nwljs ls ,dleku nwjh ij lekarj ljy js[kk,¡ gksrh gSAa 10. {ks=k js[kkvksa dh oqQN egRoiw.kZ fo'ks"krk,¡ bl izdkj gSaµ (a) vkos'keqDr fno~QLFkku esa {ks=k js[kk,¡ larr oØ gksrh gSa tks dgha ugha VwVrhaA (b) nks {ks=k js[kk,¡ ,d&nwljs dks dnkfi ugha dkV ldrhaA (c) fLFkjoS|qr {ks=k js[kk,¡ èkukos'k ls vkjaHk gksdj ½.kkos'k ij lekIr gks tkrh gSaµ ;s lao`r ik'k (can ywi) ugha cuk ldrhaA 11. oS|qr f}èkzqo ifjek.k esa leku fotkrh; nks vkos'kksa q rFkk –q, ftuosQ chp i`Fkdu 2a gks] dk ;qXe gksrk gSA blosQ f}èkzqo vk?kw.kZ lfn'k p dk ifjek.k 2qa gksrk gS rFkk ;g f}èkzqo v{k –q ls q dh fn'kk esa gksrk gSA 12. fdlh oS|rq f}èkzo q dk blosQ fuj{kh;@fo"kqorh; lery (vFkkZr blosQ v{k osQ yacor rFkk blosQ osQanz ls xqtjus okys lery) ij blosQ osQanz ls r nwjh ij fo|qr {ks=kµ −p 1 E= 4π ε o ( a + r 2 )3 / 2 2 −p ≅ , (r >> a osQ fy,) 38 4 π ε or 3 2024-25 oS|qr vkos'k rFkk {ks=k f}èkzqo v{k ij osQanz ls r nwjh ij f}èkzqo fo|qr {ks=k 2 pr E= 4 π ε 0 (r 2 − a 2 )2 2p ≅ (r >> a osQ fy,) 4π ε 0r 3 bl rF; ij è;ku nsuk pkfg, fd fdlh f}/qzo dk fo|qr {ks=k 1/r 3 ij fuHkZj gksrk gS] tcfd fdlh ¯cnq vkos'k osQ dkj.k fo|qr {ks=k 1/r 2 ij fuHkZj gksrk gSA 13. fdlh ,dleku fo|qr {ks=k E esa dksbZ oS|rq f}èkzo q ,d cy vk?kw.kZ τ dk vuqHko djrk gSA τ = p ×E ijarq fdlh usV cy dk vuqHko ugha djrkA 14. fo|qr {ks=k E dk fdlh y?kq {ks=kiQy&vo;o ∆S ls xqtjus okyk ÝyDl ∆φ = E. ∆S lfn'k {ks=kiQy&vo;o ∆S = ∆S n̂ ;gk¡ ∆S {ks=kiQy&vo;o dk ifjek.k rFkk n̂ {ks=kiQy&vo;o osQ yacor f=kT; ,dakd lfn'k gS ftls dkiQh NksVs {ks=k osQ fy, leryh; ekuk tk ldrk gSA fdlh can i`"B osQ {ks=kiQy&vo;o osQ fy,] n̂ dks ifjikVh osQ vuqlkj cfgeZ[q kh vfHkyac dh fn'kk ekuk tk ldrk gSA 15. xkml fu;e % fdlh can i`"B S ls gksdj xq”kjus okys fdlh fo|qr {ks=k dk ÝyDl ml i`"B S }kjk ifjc¼ oqQy vkos'k dk 1/ε0 xquk gksrk gSA ;g fu;e fo|qr {ks=k osQ fuèkkZj.k esa fo'ks"k :i ls rc mi;ksxh gksrk gS tcfd vkos'k forj.k esa ljy lefefr gksA (i) ,dleku jSf[kd vkos'k ?kuRo λ dk iryk vuar yackbZ dk lhèkk rkj λ E= ˆ n 2 πε0 r ;gk¡ r ¯cnq dh rkj ls yacor nwjh gS rFkk n ml ¯cnq ls xq”kjus okys rkj osQ vfHkyacor ry esa f=kT; ,dakd lfn'k gSA (ii) ,dleku i`"Bh; vkos'k ?kuRo σ dh iryh vuar lery pknj σ E= ˆ n 2ε0 ;gk¡ n̂ lery osQ vfHkyacor ik'oZ osQ nksuksa vksj cfgeZq[kh ,dakd lfn'k gSA (iii) ,dleku i`"Bh; vkos'k ?kuRo σ osQ irys xksyh; [kksy (;k dks'k) q E= rˆ (r ≥ R ) 4π ε 0 r 2 E=0 (r < R) ;gk¡ r xksyh; [kksy osQ osQanz ls ¯cnq dh nwjh rFkk R [kksy dh f=kT;k gSA [kksy dk oqQy vkos'k q gSA tgk¡ q = 4πR 2σ gSA [kksy osQ ckgj fdlh ¯cnq ij vkosf'kr [kksy osQ dkj.k fo|qr {ks=k bl izdkj gksrk gS tSls fd leLr vkos'k [kksy osQ osQanz ij gh osaQfnzr gSA ;gh ifj.kke fdlh ,dleku vk;ru vkos'k ?kuRo osQ Bksl xksys osQ fy, Hkh lR; gksrk gSA [kksy osQ vanj lHkh ¯cnqvksa ij fo|qr {ks=k 'kwU; gksrk gSA 39 2024-25 HkkSfrdh HkkSfrd jkf'k izrhd foek,¡ ek=kd fVIi.kh lfn'k {ks=kiQy-vo;o ∆S [L2] m2 ∆S = ∆S n̂ fo|qr {ks=k E [MLT A ] –3 –1 V m–1 oS|qr ÝyDl φ –3 [ML3 T A–1] Vm ∆φ = E. ∆S f}/zqo vk?kw.kZ p [LT A] Cm ½.kkos'k ls /ukos'k dh vksj fufnZ"V lfn'k vkos'k ?kuRo% jSf[kd λ [L–1 T A] C m–1 vkos'k@yackbZ i`"Bh; σ –2 [L T A] C m–2 vkos'k@{ks=kiQy vk;ru ρ [L–3 T A] C m–3 vkos'k@vk;ru fopkj.kh; fo"k; 1- vkidks ;g vk'p;Z gks ldrk gS fd ukfHkd esa izksVkWu ftu ij /ukos'k gS] lHkh ,d lkFk dls jgdj dSls lek, gq, gSaA os ,d&nwljs ls nwj D;ksa ugha mM+ tkrs\ vki ;g lh[ksaxs fd ,d rhljs izdkj dk ewy cy Hkh gS ftls izcy cy dgrs gSa vkSj ;gh cy izksVkWuksa dks ,d lkFk ck¡/s j[krk gSA ijarq nwjh dk og ifjlj ftlesa ;g cy izHkkoh gksrk gS] cgqr NksVk ~ 10-14 m gSA ;Fkk;Z :i esa ;gh ukfHkd dk lkb”k gSA lkFk gh DokaVe ;kaf=kdh ds fu;eksa ds vuqlkj bysDVªkWuksa dks izksVkWuksa ds 'kh"kZ ij vFkkZr ukfHkd ds Hkhrj cSBus dh vuqefr Hkh ugha gSA blh ls ijek.kqvksa dks mudh jpuk izkIr gS vkSj mudk izÑfr esa vfLrRo gSA 2. owQykWe cy rFkk xq#Rokd"kZ.k cy leku O;qRØe oxZ fu;e dk ikyu djrs gSaA ijarq xq#Rokd"kZ.k cy dk osQoy ,d gh fpÉ (lnSo vkd"khZ) gksrk gS] tcfd owQykWe cy osQ nksuksa fpÉ (vkd"khZ rFkk çfrd"khZ) gks ldrs gSa] ftlls fo|qr cyksa dk fujlu gks ldrk gSA ;gh dkj.k gS fd vR;ar nqcZy cy gksus ij Hkh xq#Ro cy izÑfr esa ,d izcy rFkk vf/d O;kid cy gks ldrk gSA 3. ;fn vkos'k osQ ek=kd dks owQykWe osQ fu;e }kjk ifjHkkf"kr djuk gS rks owQykWe osQ fu;e esa vkuqikfrdrk fLFkjkad k ,d p;u dk fo"k; gSA ijarq SI ek=kdksa esa fo|qr èkkjk osQ ek=kd ,sfEi;j (A) dks mlh osQ pqacdh; çHkko (,sfEi;j fu;e) }kjk ifjHkkf"kr fd;k tkrk gS rFkk vkos'k osQ ek=kd (owQykWe) dks osQoy (1C = 1A s) }kjk ifjHkkf"kr fd;k tkrk gSA bl çdj.k esa k dk eku LoSfPNd ugha gS_ ;g yxHkx 9 × 109 N m2 C−2 gSA 4. fLFkjkad k dk vR;fèkd cM+k eku] vFkkZr fo|qr çHkko dh n`f"V ls vkos'k osQ ek=kd (1C) dk cM+k vkdkj (eku) bl dkj.k ls gS D;ksafd (tSlk fd ¯cnq 3 esa mYys[k fd;k tk pqdk gS) vkos'k osQ ek=kd dks pqacdh; cyksa (fo|qrokgh rkjksa ij yxs cyksa) osQ inksa esa ifjHkkf"kr fd;k x;k gS tks fd O;kid :i ls oS|qr cyksa dh rqyuk esa cgqr nqcZy gksrs gSaA ;gh dkj.k gS fd] 1 ,sfEi;j pqacdh; çHkkoksa osQ fy, ;qfDrlaxr ek=kd gS] 1 C = 1 A s oS|qr çHkkoksa osQ fy, ,d vR;fèkd cM+k ek=kd gSA 5. vkos'k dk ;ksT;rk xq.kèkeZ dksbZ lqLi"V xq.kèkeZ ugha gSA ;g bl rF; ls lacafèkr gS fd oS|qr vkos'k ls dksbZ fn'kk lac¼ ugha gksrh] vkos'k ,d vfn'k jkf'k gSA 6. vkos'k osQoy ?kw.kZu osQ varxZr gh vfn'k (vFkok vpj@vifjorZuh;) ugha gS_ ;g vkisf{kd xfr esa Hkh funs'Z k isQz eksa osQ fy, vpj gSA ;g dFku izR;sd vfn'k ds fy, lnSo lR; ugha gSA mnkgj.kkFkZ] xfrt mQtkZ ?kw.kZu osQ varxZr vfn'k gS] ijarq ;g vkisf{kd xfr;ksa esa funsZ'k izsQeksa osQ fy, vpj ugha gS] vFkok bl n`f"V ls rks nzO;eku Hkh vpj 40 ugha gSA 2024-25 oS|qr vkos'k rFkk {ks=k 7. fdlh fo;qDr fudk; osQ oqQy vkos'k dk laj{k.k ,d ,slk xq.kèkeZ gS tks ¯cnq 6 osQ varxZr vkos'k dh vfn'k çÑfr ij fuHkZj ugha djrkA laj{k.k fdlh fn, x, funsZ'k izsQe esa le; dh vifjorZuh;rk dh vksj laosQr djrk gSA dksbZ jkf'k vfn'k gksrs gq, Hkh lajf{kr ugha gks ldrh (fdlh vçR;kLFk la?k^ esa xfrt ÅtkZ dh Hkk¡fr)A blosQ foijhr] lfn'k jkf'k dk laj{k.k Hkh gks ldrk gS (mnkgj.k osQ fy, fdlh fo;qDr fudk; dk dks.kh; laosx laj{k.k)A 8. oS|qr vkos'k dk DokaVehdj.k çÑfr dk ewy fu;e gS ftldh vHkh rd O;k[;k ugha dh tk ldh gSA jkspd rF; ;g gS fd lagfr osQ DokaVehdj.k dk dksbZ ln`'k (vuq:i) fu;e ugha gSA 9. vè;kjksi.k fl¼kar dks lqLi"V ugha ekuuk pkfg, vFkok bls lfn'kksa osQ ;ksx osQ fu;e osQ leku ugha ekuuk pkfg,A ;g fl¼kar nks ckrsa crkrk gS% fdlh vkos'k ij nwljs vkos'k osQ dkj.k cy fdUgha vU; vkos'kksa dh mifLFkfr osQ dkj.k çHkkfor ugha gksrk rFkk ;gk¡ dksbZ vfrfjDr f=k&fiaM] prq%&fiaM vkfn cy ugha gksrs tks osQoy rHkh mRiUu gksrs gSa tc nks ls vfèkd vkos'k gksaA 10. fdlh fofoDr vkos'k foU;kl osQ dkj.k fofoDr vkos'kksa dh vofLFkfr ij fo|qr {ks=k ifjHkkf"kr ugha gSA larr vk;ru vkos'k forj.k osQ fy, ;g forj.k esa fdlh Hkh ¯cnq ij ifjHkkf"kr gksrk gSA fdlh i`"Bh; vkos'k forj.k osQ fy, fo|qr {ks=k i`"B osQ vkj&ikj fofPNUu gksrk gSA 11. fdlh vkos'k foU;kl ftlesa oqQy vkos'k 'kwU; gS] osQ dkj.k lHkh fcanqvksa ij fo|qr {ks=k 'kwU; ugha gksrk_ mu nwfj;ksa osQ fy, tks vkos'k foU;kl osQ vkdkj dh rqyuk esa cM+h gSa] blosQ {ks=k esa deh 1/r 2 ls Hkh vf/d rhoz xfr ls gksrh gS] tks fd fdlh ,dy vkos'k osQ fo|qr {ks=k dh fo'ks"krk gSA ,d oS|qr f}èkqzo bl rF; dk ,d ljyre mnkgj.k gSA vH;kl 1.1 ok;q esa ,d&nwljs ls 30 cm nwjh ij j[ks nks NksVs vkosf'kr xksyksa ij Øe'k% 2 × 10−7C rFkk 3 × 10−7 C vkos'k gSaA muosQ chp fdruk cy gS\ 1.2 0.4 µ C vkos'k osQ fdlh NksVs xksys ij fdlh vU; NksVs vkosf'kr xksys osQ dkj.k ok;q esa 0.2 N cy yxrk gSA ;fn nwljs xksys ij 0.8 µC vkos'k gks rks (a) nksuksa xksyksa osQ chp fdruh nwjh gS\ (b) nwljs xksys ij igys xksys osQ dkj.k fdruk cy yxrk gS\ 1.3 tk¡p }kjk lqfuf'pr dhft, fd k e 2/G memp foekghu gSA HkkSfrd fu;rkadksa dh lkj.kh ns[kdj bl vuqikr dk eku Kkr dhft,A ;g vuqikr D;k crkrk gS\ 1.4 (a) ¶fdlh oLrq dk oS|qr vkos'k DokaVhÑr gS]¸ bl çdFku ls D;k rkRi;Z gS\ (b) LFkwy vFkok cM+s iSekus ij oS|qr vkos'kksa ls O;ogkj djrs le; ge oS|qr vkos'k osQ DokaVehdj.k dh mis{kk oSQls dj ldrs gSa\ 1.5 tc dk¡p dh NM+ dks js'ke osQ VqdM+s ls jxM+rs gSa rks nksuksa ij vkos'k vk tkrk gSA blh çdkj dh ifj?kVuk dk oLrqvksa osQ vU; ;qXeksa esa Hkh çs{k.k fd;k tkrk gSA Li"V dhft, fd ;g çs{k.k vkos'k laj{k.k fu;e ls fdl izdkj lkeatL; j[krk gSA 1.6 pkj ¯cnq vkos'k qA = 2 µC, qB = – 5 µC, qC = 2 µC rFkk qD = – 5 µC, 10 cm Hkqtk osQ fdlh oxZ ABCD osQ 'kh"kks± ij vofLFkr gSAa oxZ osQ osQæa ij j[ks 1 µC vkos'k ij yxus okyk cy fdruk gS\ 1.7 (a) fLFkjoS|qr {ks=k js[kk ,d larr oØ gksrh gS vFkkZr dksbZ {ks=k js[kk ,dk,d ugha VwV ldrhA D;ksa\ (b) Li"V dhft, fd nks {ks=k js[kk,¡ dHkh Hkh ,d&nwljs dk çfrPNsnu D;ksa ugha djrha\ 41 2024-25 HkkSfrdh 1.8 nks ¯cnq vkos'k qA = 3 µC rFkk qB = – 3 µ C fuokZr esa ,d&nwljs ls 20 cm nwjh ij fLFkr gSAa (a) nksuksa vkos'kksa dks feykus okyh js[kk AB osQ eè; ¯cnq O ij fo|qr {ks=k fdruk gS\ (b) ;fn 1.5 × 10−9 C ifjek.k dk dksbZ ½.kkRed ijh{k.k vkos'k bl ¯cnq ij j[kk tk, rks ;g ijh{k.k vkos'k fdrus cy dk vuqHko djsxk\ 1.9 fdlh fudk; esa nks vkos'k qA = 2.5 × 10−7 C rFkk qB = –2.5 × 10−7 C Øe'k% nks ¯cnqvksa A : (0, 0, –15 cm) rFkk B : (0, 0, +15 cm) ij vofLFkr gSaA fudk; dk oqQy vkos'k rFkk oS|qr f}èkzqo vk?kw.kZ D;k gS\ 1.10 4 × 10−9 C m f}èkqzo vk?kw.kZ dk dksbZ oS|qr f}èkqzo 5 × 104 N C−1 ifjek.k osQ fdlh ,dleku fo|qr {ks=k dh fn'kk ls 30° ij lajsf[kr gSA f}èkqzo ij dk;Zjr cy vk?kw.kZ dk ifjek.k ifjdfyr dhft,A 1.11 Åu ls jxM+s tkus ij dksbZ ikWyhFkhu dk VqdM+k 3 × 10−7 C osQ ½.kkos'k ls vkosf'kr ik;k x;kA (a) LFkkukarfjr (fdl inkFkZ ls fdl inkFkZ esa) bysDVªkWuksa dh la[;k vkdfyr dhft,A (b) D;k Åu ls ikWyhFkhu esa lagfr dk LFkkukarj.k Hkh gksrk gS\ 1.12 (a) nks fo|qrjksèkh vkosf'kr rk¡cs osQ xksyksa A rFkk B osQ osaQæksa osQ chp dh nwjh 50 cm gSA ;fn nksuksa xksyksa ij i`Fkd&i`Fkd vkos'k 6.5 × 10−7 C gSa] rks buesa ikjLifjd fLFkjoS|qr çfrd"kZ.k cy fdruk gS\ xksyksa osQ chp dh nwjh dh rqyuk esa xksyksa A rFkk B dh f=kT;k,¡ ux.; gSaA (b) ;fn çR;sd xksys ij vkos'k dh ek=kk nks xquh rFkk xksyksa osQ chp dh nwjh vkèkh dj nh tk, rks çR;sd xksys ij fdruk cy yxsxk\ 1.13 fp=k 1.30 esa fdlh ,dleku fLFkjoS|rq {ks=k eas rhu vkosf'kr d.kksa osQ iFkfpÉ (tracks) n'kkZ, x, gSaA rhuksa vkos'kksa osQ fpÉ fyf[k,A buesa ls fdl d.k dk vkos'k&lagfr vuqikr (q/m) vfèkdre gS\ fp=k 1.30 1.14 ,dleku fo|qr {ks=k E = 3 × 103 î N/C ij fopkj dhft,A (a) bl {ks=k dk 10 cm Hkqtk osQ oxZ osQ ml ik'oZ ls ftldk ry yz ry osQ lekarj gS] xqtjus okyk ÝyDl D;k gS\ (b) blh oxZ ls xqtjus okyk ÝyDl fdruk gS ;fn blosQ ry dk vfHkyac x -v{k ls 60° dk dks.k cukrk gS\ 1.15 vH;kl 1.14 osQ ,dleku fo|qr {ks=k dk 20 cm Hkqtk osQ fdlh ?ku ls (tks bl çdkj vfHkfoU;kflr gS fd mlosQ iQyd funsZ'kkad ryksa osQ lekarj gSa) fdruk usV ÝyDl xqtjsxk\ 1.16 fdlh dkys ckWDl osQ i`"B ij fo|qr {ks=k dh lkoèkkuhiwoZd yh xbZ eki ;g laosQr nsrh gS fd ckWDl osQ i`"B ls xqtjus okyk usV ÝyDl 8.0 × 103 N m2/C gSA (a) ckWDl osQ Hkhrj usV vkos'k fdruk gS\ (b) ;fn ckWDl osQ i`"B ls usV cfgeZq[kh ÝyDl 'kwU; gS rks D;k vki ;g fu"d"kZ fudkysaxs fd ckWDl osQ Hkhrj dksbZ vkos'k ugha gS\ D;ksa] vFkok D;ksa ugha\ 1.17 fp=k 1.31 esa n'kkZ, vuqlkj 10 cm Hkqtk osQ fdlh oxZ osQ osQaæ ls Bhd 5 cm Å¡pkbZ ij dksbZ +10 µ C vkos'k j[kk gSA bl oxZ ls xqtjus okys oS|qr ÝyDl dk ifjek.k D;k gS\ (laosQr % 42 oxZ dks 10 cm fdukjs osQ fdlh ?ku dk ,d iQyd ekfu,A) 2024-25 oS|qr vkos'k rFkk {ks=k fp=k 1.31 1.18 2.0 µC dk dksbZ ¯cnq vkos'k fdlh fdukjs ij 9.0 cm fdukjs okys fdlh ?kuh; xkmlh; i`"B osQ osQaæ ij fLFkr gSA i`"B ls xqtjus okyk usV ÝyDl D;k gS\ 1.19 fdlh ¯cnq vkos'k osQ dkj.k ml ¯cnq dks osQæa ekudj [khaps x, 10 cm f=kT;k osQ xksyh; xkmlh; i`"B ij oS|qr ÝyDl –1.0 × 103 N m2/CA (a) ;fn xkmlh; i`"B dh f=kT;k nks xquh dj nh tk, rks i`"B ls fdruk ÝyDl xqtjsxk\ (b) ¯cnq vkos'k dk eku D;k gS\ 1.20 10 cm f=kT;k osQ pkyd xksys ij vKkr ifj.kke dk vkos'k gSA ;fn xksys osQ osaQæ ls 20 cm nwjh ij fo|qr {ks=k 1.5 × 103 N/C f=kT;r% vareZq[kh (radially inward) gS rks xksys ij usV vkos'k fdruk gS\ 1.21 2.4 m O;kl osQ fdlh ,dleku vkosf'kr pkyd xksys dk i`"Bh; vkos'k ?kuRo 80.0 µC/m2 gSA (a) xksys ij vkos'k Kkr dhft,A (b) xksys osQ i`"B ls fuxZr oqQy oS|qr ÝyDl D;k gS\ 1.22 dksbZ vuar jSf[kd vkos'k 2 cm nwjh ij 9 × 104 N C−1 fo|qr {ks=k mRiUu djrk gSA jSf[kd vkos'k ?kuRo Kkr dhft,A 1.23 nks cM+h] iryh èkkrq dh IysVas ,d&nwljs osQ lekukarj ,oa fudV gSAa buosQ Hkhrjh iQydksa ij] IysVksa osQ i`"Bh; vkos'k ?kuRoksa osQ fpÉ foijhr gSa rFkk budk ifjek.k 17.0 × 10–22 C/m2 gSA (a) igyh IysV osQ cká {ks=k es]a (b) nwljh IysV osQ cká {ks=k es]a rFkk (c) IysVksa osQ chp esa fo|qr {ks=k E dk ifjek.k ifjdfyr dhft,A 43 2024-25
